If \ mathbf { x } is a nonnegative integral vector indexed by the non-sink vertices, then starting with a configuration z and toppling each vertex v a total of \ mathbf { x } ( v ) times yields the configuration z-\ mathbf { x } \ Delta '.
2.
More formally, if \ mathbf { u } is a vector such that \ mathbf { u } ( v ) is the number of times the vertex v topples during the stabilization ( via the toppling of unstable vertices ) of a chip configuration z, and \ mathbf { n } is an integral vector ( not necessarily non-negative ) such that z-\ mathbf { n } \ Delta'is stable, then \ mathbf { u } ( v ) \ leq \ mathbf { n } ( v ) for all vertices v.
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